ON THE REPRESENTATION OF OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS

Transcription

1 SIAM J. NUMER. ANAL. c 1992 Society for Industria Appied Mathematics Vo. 6, No. 6, pp , December ON THE REPRESENTATION OF OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS G. BEYLKIN Abstract. This paper describes exact expicit representations of the differentia operators, d n /dx n, n = 1, 2,, in orthonorma bases of compacty supported waveets as we as the representations of the Hibert transform fractiona derivatives. The method of computing these representations is directy appicabe to mutidimensiona convoution operators. Aso, sparse representations of shift operators in orthonorma bases of compacty supported waveets are discussed a fast agorithm requiring O(N og N) operations for computing the waveet coefficients of a N circuant shifts of a vector of the ength N = 2 n is constructed. As an exampe of an appication of this agorithm, it is shown that the storage requirements of the fast agorithm for appying the stard form of a pseudodifferentia operator to a vector (see [G. Beykin, R. R. Coifman, V. Rokhin, Comm. Pure. App. Math., 44 (1991), pp ]) may be reduced from O(N) to O(og 2 N) significant entries. Key words. waveets, differentia operators, Hibert transform, fractiona derivatives, pseudodifferentia operators, shift operators, numerica agorithms AMS(MOS) subject cassifications. 65D99, 35S99, 65R10, 44A15 1. Introduction. In [1] Daubechies introduced compacty supported waveets which proved to be very usefu in numerica anaysis [2]. In this paper we find exact expicit representations of severa basic operators (derivatives, Hibert transform, shifts, etc.) in orthonorma bases of compacty supported waveets. We aso present an O(N og N) agorithm for computing the waveet coefficients of a N circuant shifts of a vector of the ength N = 2 n. Throughout this paper we ony compute the nonstard forms of operators since it is a simpe matter to obtain a stard form from the nonstard form [2]. Meyer [3], foowing [2], considered severa exampes of nonstard forms of basic operators from a genera point of view. It is possibe, however, to compute the nonstard forms of many important operators expicity. First, we expicity compute the nonstard form of the operator d/dx. The set of coefficients that defines a nonzero entries of the nonstard form appears as the soution to a system of inear agebraic equations. This system, in turn, arises as a consequence of the recursive definition of the waveet bases. The operator d n /dx n is treated simiary to d/dx. The computation of the nonstard forms of many other operators reduces to soving a simpe system of inear agebraic equations. Among such operators are fractiona derivatives, Hibert Riesz transforms, other operators for which anaytic expressions are avaiabe. For convoution operators, there are significant simpifications in computing the nonstard form since the vanishing moments of the autocorreation function of the scaing function simpify the quadrature formuas. Moreover, by soving a system of inear agebraic equations combined with the asymptotics of waveet coefficients, we arrive at an effective method for computing the nonstard form of convoution operators. As exampes, we compute the nonstard forms of the Hibert transform fractiona derivatives. The generaization of this method for mutidimensiona convoution operators is straightforward. Received by the editors August 6, 1990; accepted for pubication (in revised form) October 21, Schumberger-Do Research, Od Quarry Road, Ridgefied, Connecticut Present address, Program in Appied Mathematics, University of Coorado at Bouder, Bouder, Coorado

2 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1717 Second, we compute the nonstard form of the shift operator. This operator is important in practica appications of waveets because the waveet coefficients are not shift invariant. Since the nonstard stard forms of this operator are sparse easy to compute, knowing these representations compensates for the ack of shift invariance. The waveet expansion of shifts of vectors or of matrices may be obtained by appying the shift operator directy to the coefficients of the origina expansion. The coefficients for the shift operators may be stored in advance used as needed. It is cear, however, that the particuar manner in which sparseness of the shift operator may be expoited depends on the appication may be ess straightforward than is indicated above. We present an exampe of such an appication in numerica anaysis. Observing that there are ony N og 2 N distinct waveet coefficients in the decomposition of a N circuant shifts of a vector of the ength N = 2 n, we construct an O(N og N) agorithm for computing a of these coefficients. Using this agorithm, we show that the storage requirements of the fast agorithm for appying the stard form of a pseudodifferentia operator to a vector [2] may be reduced from O(N og N) to O(og 2 N) significant entries. 2. Compacty supported waveets. In this section, we briefy review the orthonorma bases of compacty supported waveets set our notation. For the detais we refer to [1]. The orthonorma basis of compacty supported waveets of L 2 (R) is formed by the diation transation of a singe function ψ(x), (2.1) ψ j,k (x) = 2 j/2 ψ(2 j x k), where j, k Z. The function ψ(x) has a companion, the scaing function ϕ(x), these functions satisfy the foowing reations: (2.2) ϕ(x) = 2 h k ϕ(2x k), k=0 (2.3) ψ(x) = 2 g k ϕ(2x k), k=0 where (2.4) g k = ( 1) k h L k 1, k = 0,, L 1, (2.5) ϕ(x)dx = 1. In addition, the function ψ has M vanishing moments (2.6) ψ(x)x m dx = 0, m = 0,, M 1.

3 1718 G. BEYLKIN The number L of coefficients in (2.2) (2.3) is reated to the number of vanishing moments M, for the waveets in [1], L = 2M. If additiona conditions are imposed (see [2] for an exampe), then the reation might be different, but L is aways even. The waveet basis induces a mutiresoution anaysis on L 2 (R) [4], [5], i.e., the decomposition of the Hibert space L 2 (R) into a chain of cosed subspaces (2.7) V 2 V 1 V 0 V 1 V 2 such that (2.8) V j = {0}, j Z V j = L 2 (R). j Z By defining W j as an orthogona compement of V j in V j 1, (2.9) V j 1 = V j W j, the space L 2 (R) is represented as a direct sum (2.10) L 2 (R) = j Z W j. On each fixed scae j, the waveets {ψ j,k (x)} k Z form an orthonorma basis of W j the functions {ϕ j,k (x) = 2 j/2 ϕ(2 j x k)} k Z form an orthonorma basis of V j. The coefficients H = {h k } k= k=0 G = {g k } k= k=0 in (2.2) (2.3) are quadrature mirror fiters. Once the fiter H has been chosen, it competey determines the functions ψ ϕ. Let us define the 2π-periodic function (2.11) k= m 0 (ξ) = 2 1/2 k=0 h k e ikξ where {h k } k= k=0 are the coefficients of the fiter H. The function m 0 (ξ) satisfies the equation (2.12) m 0 (ξ) 2 + m 0 (ξ + π) 2 = 1. The foowing emma characterizes trigonometric poynomia soutions of (2.12) which correspond to the orthonorma bases of compacty supported waveets with vanishing moments. Lemma 1 (Daubechies [1]). Any trigonometric poynomia soution m 0 (ξ) of (2.12) is of the form (2.13) m 0 (ξ) = [ 1 2(1 + e iξ ) ] M Q(e iξ ), where M 1 is the number of vanishing moments, where Q is a poynomia such that (2.14) Q(e iξ ) 2 = P (sin 2 1 2ξ) + sin 2M ( 2ξ) 1 R( 1 2 cos ξ),

4 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1719 where (2.15) P (y) = k=m 1 k=0 R is an odd poynomia such that ( M 1 + k k ) y k, (2.16) 0 P (y) + y M R( 1 2 y) for 0 y 1, (2.17) [ sup P (y) + y M R( 1 2 y) ] < 2 2(M 1). 0 y 1 3. The operator d/dx in waveet bases. In this section we construct the nonstard form of the operator d/dx. The nonstard form [2] is a representation of an operator T as a chain of tripets (3.1) T = {A j, B j, Γ j } j Z acting on the subspaces V j W j, (3.2) (3.3) (3.4) A j : W j W j, B j : V j W j, Γ j : W j V j. The operators {A j, B j, Γ j } j Z are defined as A j = Q j T Q j, B j = Q j T P j, Γ j = P j T Q j, where P j is the projection operator on the subspace V j Q j = P j 1 P j is the projection operator on the subspace W j. The matrix eements α j i, βj i, γj i of A j, B j, Γ j, r j i of T j = P j T P j, i,, j Z, for the operator d/dx are easiy computed as (3.5) (3.6) (3.7) (3.8) where (3.9) α j i = 2 j β j i = 2 j γ j i = 2 j r j i = 2 j ψ(2 j x i) ψ (2 j x ) 2 j dx = 2 j α i, ψ(2 j x i) ϕ (2 j x ) 2 j dx = 2 j β i, ϕ(2 j x i) ψ (2 j x ) 2 j dx = 2 j γ i, ϕ(2 j x i) ϕ (2 j x ) 2 j dx = 2 j r i, α = ψ(x ) d ψ(x) dx, dx

5 1720 G. BEYLKIN (3.10) β = ψ(x ) d ϕ(x) dx, dx (3.11) (3.12) γ = r = Moreover, using (2.2) (2.3) we have ϕ(x ) d ψ(x) dx. dx ϕ(x ) d ϕ(x) dx. dx (3.13) α i = 2 k=0 k =0 g k g k r 2i+k k, (3.14) (3.15) β i = 2 γ i = 2 k=0 k =0 k=0 k =0 g k h k r 2i+k k, h k g k r 2i+k k,, therefore, the representation of d/dx is competey determined by r in (3.12) or, in other words, by the representation of d/dx on the subspace V 0. Rewriting (3.12) in terms of ˆϕ(ξ), where (3.16) we obtain (3.17) ˆϕ(ξ) = 1 ϕ(x) e ixξ dx, 2π r = ( iξ) ˆϕ(ξ) 2 e iξ dξ. In order to compute the coefficients r we first note that any trigonometric poynomia m 0 (ξ) satisfying (2.12) is such that (3.18) L/2 m 0 (ξ) 2 = a 2k 1 cos(2k 1)ξ, 2 k=1 where a n are the autocorreation coefficients of H = {h k } k= k=0, (3.19) n a n = 2 h i h i+n, n = 1,, L 1. i=0 The autocorreation coefficients a n with even indices are zero, (3.20) a 2k = 0, k = 1,, L/2 1.

6 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1721 To prove this assertion we compute m 0 (ξ) 2 using (2.11) obtain (3.21) m 0 (ξ) 2 = a n cos nξ, 2 n=1 where a n are given in (3.19). Computing m 0 (ξ + π) 2, we have (3.22) L/2 m 0 (ξ + π) 2 = a 2k 1 cos(2k 1)ξ k=1 Combining (3.21) (3.22) to satisfy (2.12), we obtain (3.23) L/2 1 k=1 a 2k cos 2kξ = 0, L/2 1 k=1 a 2k cos 2kξ. hence, (3.20) (3.18). (See aso Remark 6 about vanishing moments of a 2k 1.) We prove the foowing: Proposition 1. (1) If the integras in (3.12) or (3.17) exist, then the coefficients r in (3.12) satisfy the foowing system of inear agebraic equations: (3.24) r = 2 r L/2 a 2k 1 (r 2 2k+1 + r 2+2k 1 ), 2 k=1 (3.25) r = 1, where the coefficients a 2k 1 are given in (3.19). (2) If M 2, then equations (3.24) (3.25) have a unique soution with a finite number of nonzero r, namey, r 0 for L + 2 L 2 (3.26) r = r. Remark 1. If M = 1, then equations (3.24) (3.25) have a unique soution but the integras in (3.12) or (3.17) may not be absoutey convergent. Let us consider Exampe 3.2 of [1], where L = 4 1/2 ν(ν 1) h 0 = 2 ν 2 + 1, h 1 = 2 1/2 1 ν ν 2 + 1, h 2 = 2 1/2 ν + 1 ν 2 + 1, h 1/2 ν(ν + 1) 3 = 2 ν 2 + 1, where ν is an arbitrary rea number. We have a 1 = 1 + 3ν2 (ν 2 + 1) 2, a 3 = ν2 (ν 2 1) (ν 2 + 1) 2, r 1 = (1 + ν2 ) 2 2(3ν 4 + 1), r 2 = ν2 (1 ν 2 ) 2(3ν 4 + 1).

7 1722 G. BEYLKIN The parameter ν can be chosen so that the Fourier transform ˆϕ(ξ) does not have the sufficient decay to insure the absoute convergence of the integra (3.17). For the Haar basis (h 1 = h 2 = 2 1/2 ), a 1 = 1 r 1 = 1 2, thus, we obtain the simpest finite difference operator ( 1 2, 0, 1 2 ). In this case the function ϕ is not continuous ˆϕ(ξ) = 1 sin 1 2 ξ 2π 1 2 ξ e i 1 2 ξ, so that the integra in (3.17) is not absoutey convergent. Proof of Proposition 1. Using (2.2) for both ϕ(x ) obtain d dx ϕ(x) in (3.12) we (3.27) r i = 2 k=0 =0 h k h ϕ(2x 2i k) ϕ (2x ) 2 dx hence, (3.28) r i = 2 k=0 =0 h k h r 2i+k. Substituting = k m, we rewrite (3.28) as (3.29) r i = 2 k=0 k L+1 m=k h k h k m r 2i+m. Changing the order of summation in (3.29) using the fact that arrive at k=0 h2 k = 1, we (3.30) r = 2r 2 + a n (r 2 n + r 2+n ), Z, n=1 where a n are given in (3.19). Using (3.20), we obtain (3.24) from (3.30). In order to obtain (3.25) we use the foowing reation: (3.31) where =+ = =m m ϕ(x ) = x m + =1 ( 1) ( m ) M ϕ x m, (3.32) M ϕ = ϕ(x) x dx, where = 1,, m, are the moments of the function ϕ(x). We note that (3.31) is we known if a moments (3.32) are zero. The genera statement foows simpy on taking Fourier transforms using Leibniz s rue. Using (3.12) (3.31) with m = 1, we obtain (3.25). If M 2, then (3.33) ˆϕ(ξ) 2 ξ C(1 + ξ ) 1 ɛ,

8 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1723 where ɛ > 0, hence, the integra in (3.17) is absoutey convergent. This assertion foows from Lemma 3.2 of [1], where it is shown that (3.34) ˆϕ(ξ) C(1 + ξ ) M+og 2 B, where B = sup Q(e iξ ). ξ R Due to condition (2.17), we have og 2 B = M 1 ɛ with some ɛ > 0. The existence of the soution of the system of equations (3.24) (3.25) foows from the existence of the integra in (3.17). Since the scaing function ϕ has a compact support there are ony a finite number of nonzero coefficients r. The specific interva L+2 L 2 is obtained by the direct examination of (3.24). Let us show now that (3.35) r = 0. Mutipying (3.24) by e iξ summing over, we obtain ˆr(ξ) = 2 ˆr even (ξ/2) + 1 L/2 2 ˆr (3.36) odd(ξ/2) a 2k 1 (e i(2k 1)ξ/2 + e i(2k 1)ξ/2 ), where k=1 (3.37) ˆr(ξ) = r e iξ, (3.38) ˆr even (ξ/2) = r 2 e iξ, (3.39) ˆr odd (ξ/2) = r 2+1 e i(2+1) ξ/2. Noticing that (3.40) 2 ˆr even (ξ/2) = ˆr(ξ/2) + ˆr(ξ/2 + π) (3.41) 2 ˆr odd (ξ/2) = ˆr(ξ/2) ˆr(ξ/2 + π), using (3.18), we obtain from (3.36) (3.42) ˆr(ξ) = [ ˆr(ξ/2) + ˆr(ξ/2 + π) + (2 m 0 (ξ/2) 2 1)(ˆr(ξ/2) ˆr(ξ/2 + π)) ]. Finay, using (2.12) we arrive at (3.43) ˆr(ξ) = 2( m 0 (ξ/2) 2 ˆr(ξ/2) + m 0 (ξ/2) + π) 2 ˆr(ξ/2) + π)).

9 1724 G. BEYLKIN Setting ξ = 0 in (3.43), we obtain ˆr(0) = 2ˆr(0) thus, (3.35). Uniqueness of the soution of (3.24) (3.25) foows from the uniqueness of the representation of d/dx. Given the soution r of (3.24) (3.25) we consider the operator T j defined by these coefficients on the subspace V j appy it to a sufficienty smooth function f. Since r j = 2 j r (3.8), we have ( (3.44) (T j f)(x) = k Z 2 j r f j,k ) ϕ j,k (x), where (3.45) f j,k = 2 j/2 f(x) ϕ(2 j x k + ) dx. Rewriting (3.45) (3.46) f j,k = 2 j/2 f(x 2 j ) ϕ(2 j x k) dx, exping f(x 2 j ) in the Tayor series at the point x, we have (3.47) f j,k = f(x) ϕ j,k (x) dx 2 j +2 2j 2 2 f ( x) ϕ j,k (x) dx, f (x) ϕ j,k (x) dx where x = x(x, x 2 j ) x x 2 j. Substituting (3.47) in (3.44) using (3.35) (3.25), we obtain (3.48) (T j f)(x) = k Z ( +2 j k Z ) f (x) ϕ j,k (x) dx ϕ j,k (x) ( 1 2 ) + r 2 f ( x) ϕ j,k (x) dx ϕ j,k (x). It is cear that as j, operators T j d/dx coincide on smooth functions. Using stard arguments it is easy to prove that T = d/dx hence, the soution to (3.24) (3.25) is unique. The reation (3.26) foows now from (3.17). Remark 2. We note that expressions (3.13) (3.14) for α β (γ = β ) may be simpified by changing the order of summation in (3.13) (3.14) by introducing the correation coefficients 2 n i=0 g i h i+n, 2 n i=0 h i g i+n 2 n i=0 g i g i+n. The expression for α is especiay simpe: α = 4r 2 r. Exampes. For exampes we wi use Daubechies waveets constructed in [1]. First, et us compute the coefficients a 2k 1, k = 1,, M, where M is the number of vanishing moments L = 2M. Using reation (4.22) of [1], (3.49) m 0 (ξ) 2 = 1 (2M 1)! [(M 1)!] 2 2 2M 1 ξ 0 sin 2M 1 ξ dξ,

10 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1725 we find, by computing ξ 0 sin2m 1 ξ dξ, that (3.50) where (3.51) m 0 (ξ) 2 = C M M m=1 ( 1) m 1 cos(2m 1)ξ (M m)! (M + m 1)! (2m 1), [ ] 2 (2M 1)! C M = (M 1)! 4 M 1. Thus, by comparing (3.50) (3.18), we have (3.52) a 2m 1 = ( 1) m 1 C M, where m = 1,, M. (M m)! (M + m 1)! (2m 1) The coefficients r e are rationa numbers since they are soutions of a inear system with rationa coefficients (a 2m 1 in (3.52) are rationa by construction). We note that the coefficients r are the same for a Daubechies waveets with a fixed number of vanishing moments M, notwithsting the fact that there are severa different waveet bases for a given M (depending on the choice of the roots of poynomias in the construction described in [1]). Soving the equations of Proposition 1, we present the resuts for Daubechies waveets with M = 2, 3, 4, 5, M = 2. a 1 = 9 8, a 3 = 1 8, r 1 = 2 3, r 2 = The coefficients ( 1/12, 2/3, 0, 2/3, 1/12) of this exampe coincide with one of the stard choices of coefficients for numerica differentiation. 2. M = M = 4. a 1 = 75 64, a 3 = , a 5 = 3 128, r 1 = , r 2 = , r 3 = , r 4 = a 1 = , a 3 = , a 5 = , a 7 = , r 1 = , r 2 = , r 3 = , r 4 = , r 5 = , r 1 6 =

11 1726 G. BEYLKIN 4. M = 5. a 1 = , a 3 = , a 5 = , a 7 = , a 9 = , r 1 = , r 2 = , r 3 = , r 4 = , r 5 = , r 6 = , 2048 r 7 = , r 5 8 = M = 6. a 1 = , a 3 = , a 5 = , a 7 = , a 9 = , a 11 = , r 1 = , r 2 = , r 3 = , r 4 = , r 5 = , r = , r 7 = r 10 = , r 8 = , r 9 = , Iterative agorithm for computing the coefficients r. We aso use an iterative agorithm as a way of soving equations (3.24) (3.25). We start with r 1 = 0.5 r 1 = 0.5, iterate using (3.24) to recompute r. Using (3.43), it is easy to verify that (3.25) (3.26) are satisfied due to the choice of initiaization. Tabe 1 was computed using this agorithm for Daubechies waveets with M = 5, 6, 7, 8, 9. It ony dispays the coefficients {r } L 2 =1 since r = r r 0 = The operator d n /dx n in waveet bases. Again, as in the case with the operator d/dx, the nonstard form of the operator d n /dx n is competey determined by its representation on the subspace V 0, i.e., by the coefficients (4.1) or, aternativey, (4.2) r (n) = r (n) = ϕ(x ) dn ϕ(x) dx, Z, dxn ( iξ) n ˆϕ(ξ) 2 e iξ dξ if the integras in (4.1) or (4.2) exist (see aso Remark 3).

12 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1727 Tabe 1 The coefficients {r } =L 2 for Daubechies waveets, where L = 2M M = 5,, 9. =1 Coefficients Coefficients r r M = M = E E E E E E E E E E E E E-08 M = E E E E E E E E E-05 M = E E E E E E-03 M = E E E E E E E E E E E E E E E E E E E E-16

13 1728 G. BEYLKIN Proposition 2. (1) If the integras in (4.1) or (4.2) exist, then the coefficients r (n), Z satisfy the foowing system of inear agebraic equations: (4.3) (4.4) r (n) = 2 n r L/2 2 k=1 n r (n) a 2k 1 (r (n) 2 2k+1 + r(n) = ( 1) n n!, 2+2k 1 ), where a 2k 1 are given in (3.19). (2) Let M (n + 1)/2, where M is the number of vanishing moments in (2.6). If the integras in (4.1) or (4.2) exist, then the equations (4.3) (4.4) have a unique soution with a finite number of nonzero coefficients r (n) 0 for L + 2 L 2, such that for even n r (n), namey, (4.5) (4.6) (4.7) (4.8) for odd n r (n) = r (n), 2ñ r (n) = 0, ñ = 1,, n/2 1, r (n) r (n) = 0, = r (n), (4.9) 2ñ 1 r (n) = 0, ñ = 1,, (n 1)/2. The proof of Proposition 2 is competey anaogous to that of Proposition 1. Remark 3. The inear system in Proposition 2 may have a unique soution whereas integras (4.1) (4.2) are not absoutey convergent. A case in point is the Daubechies waveet with M = 2. The representation of the first derivative in this basis is described in the previous section. Equations (4.3) (4.4) do not have a soution for the second derivative n = 2. However, the system of equations (4.3) (4.4) has a soution for the third derivative n = 3. We have a 1 = 9 8, a 3 = 1 8, r 2 = 1 2, r 1 = 1, r 0 = 0, r 1 = 1, r 2 = 1 2.

14 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1729 The set of coefficients ( 1/2, 1, 0, 1, 1/2) is one of the stard choices of finite difference coefficients for the third derivative. We note that among the waveets with L = 4, the waveets with two vanishing moments M = 2 do not have the best Höder exponent (see [6]), but the representation of the third derivative exists ony if the number of vanishing moments M = 2. Remark 4. Let us derive an equation generaizing to (3.43) for d n /dx n directy from (4.2). We rewrite (4.2) as (4.10) 2π r (n) = 0 ˆϕ(ξ + 2πk) 2 ( i) n (ξ + 2πk) n e iξ dξ. k Z Therefore, (4.11) ˆr(ξ) = k Z ˆϕ(ξ + 2πk) 2 ( i) n (ξ + 2πk) n, where (4.12) ˆr(ξ) = r (n) e iξ. Substituting the reation (4.13) ˆϕ(ξ) = m 0 (ξ/2) ˆϕ(ξ/2) into the right-h side of (4.11), summing separatey over even odd indices in (4.11), we arrive at (4.14) ˆr(ξ) = 2 n ( m 0 (ξ/2) 2 ˆr(ξ/2) + m 0 (ξ/2) + π) 2 ˆr(ξ/2) + π)). By considering the operator M 0 defined on 2π-periodic functions, (4.15) (M 0 f)(ξ) = m 0 (ξ/2) 2 f(ξ/2) + m 0 (ξ/2) + π) 2 f(ξ/2) + π), we rewrite (4.14) as (4.16) M 0ˆr = 2 nˆr. Thus, ˆr is an eigenvector of the operator M 0 corresponding to the eigenvaue 2 n, therefore, finding the representation of the derivatives in the waveet basis is equivaent to finding trigonometric poynomia soutions of (4.16) vice versa. (The operator M 0 is aso introduced in [7] [8], where the probem (4.16) with eigenvaue 1 is considered.) Remark 5. Whie theoreticay it is we understood that the derivative operators (or, more generay, operators with homogeneous symbos) have an expicit diagona preconditioner in waveet bases, the numerica evidence iustrating this fact is of interest, since it represents one of the advantages of computing in the waveet bases. If an operator has a nu space (the actua nu space or a nu space for a given accuracy), then by the condition number we underst the ratio of the argest singuar vaue to the smaest singuar vaue above the threshod of accuracy. Thus, we incude the situation where the operator may be preconditioned ony on a subspace. We note that the preconditioning described here addresses the probem of i conditioning due

15 1730 G. BEYLKIN ony to the unfavorabe homogeneity of the symbo does not affect i conditioning due to other causes. For periodized derivative operators the bound on the condition number depends ony on the particuar choice of the waveet basis. After appying such a preconditioner, the condition number κ p of the operator is uniformy bounded with respect to the size of the matrix. We reca that the condition number contros the rate of convergence of a number of iterative agorithms; for exampe, the number of iterations of the conjugate gradient method is O( κ p ). Thus, this remark impies a competey new outook on a number of numerica methods, a topic we wi address esewhere. We present here two tabes iustrating such preconditioning appied to the stard form of the second derivative (see [2] on how to compute the stard form from the nonstard form). In the foowing exampes the stard form of the periodized second derivative D 2 of size N N, where N = 2 n, is preconditioned by the diagona matrix P, D p 2 = P D 2P where P i = δ i 2 j, 1 j n, where j is chosen depending on i, so that N N/2 j i, N N/2 j, P NN = 2 n. Tabes 2 3 compare the origina condition number κ of D 2 κ p of D p 2. Tabe 2 Condition numbers of the matrix of periodized second derivative (with without preconditioning) in the basis of Daubechies waveets with three vanishing moments M = 3. N κ κ p E E E E E E E E E E Convoution operators. For convoution operators, the computation of the nonstard form is consideraby simper than in the genera case [2]. We wi demonstrate that the quadrature formuas for representing kernes of convoution operators on V 0 (see, e.g., Appendix B of [2]) are of the simpest form due to the fact that the moments of the autocorreation function of the scaing function ϕ vanish. Moreover, by combining the asymptotics of waveet coefficients with the system of inear agebraic equations (simiar to those in previous sections), we arrive at an effective method for computing representations of convoution operators. This method is especiay simpe if the symbo of the operator is homogeneous of some degree. Let us assume that the matrix t (j 1) i (i, Z) represents the operator P j 1 T P j 1 on the subspace V j 1. To compute the matrix representation of P j T P j, we have the foowing formua (3.26) of [2]: (5.1) t (j) = k=0 m=0 h k h m t (j 1) 2+k m,

16 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1731 Tabe 3 Condition numbers of the matrix of periodized second derivative (with without preconditioning) in the basis of Daubechies waveets with six vanishing moments M = 6. N κ κ p E E E E E E E E E E+01 which easiy reduces to (5.2) t (j) = t (j 1) L/2 2 k=0 a 2k 1 (t (j 1) 2 2k+1 + t(j 1) 2+2k 1 ), where the coefficients a 2k 1 are given in (3.19). We aso have (5.3) t (j) = K(x y) ϕ j,0 (y) ϕ j, (x) dxdy, by changing the order of integration, we obtain (5.4) t (j) = 2 j K(2 j ( y)) Φ(y) dy, where Φ is the autocorreation function of the scaing function ϕ, (5.5) Φ(y) = ϕ(x) ϕ(x y) dx. Let us verify that (5.6) Φ(y)dy = 1 (5.7) Ceary, we have (5.8) M m Φ = y m Φ(y)dy = 0 for 1 m 2M 1. M m Φ = [( ) m ] 1 i ξ ˆϕ(ξ) 2. ξ=0

17 1732 G. BEYLKIN Using (5.8) the identity ˆϕ(ξ) = ˆϕ(ξ/2)m 0 (ξ/2) (see [1]), it is cear that (5.7) hods provided that [( ) m ] 1 (5.9) i ξ m 0 (ξ) 2 = 0 for 1 m 2M 1, ξ=0 or (due to (2.12)) [( ) m ] 1 (5.10) i ξ m 0 (ξ + π) 2 ξ=0 = 0 for 0 m 2M 1. But formua (5.10) foows from the expicit representation in (2.13). Remark 6. Equations (5.9) (3.21) aso impy that even moments of the coefficients a 2k 1 from (3.19) vanish, namey, (5.11) k=l/2 k=1 a 2k 1 (2k 1) 2m = 0 for 1 m M 1. Since the moments of the function Φ vanish equation (5.4) eads to a one-point quadrature formua for computing the representation of convoution operators on the finest scae. This formua is obtained in exacty the same manner as for the specia choice of the waveet basis described in [2, eqns. (3.8) (3.12)], where the shifted moments of the function ϕ vanish; we refer to this paper for the detais. Here we introduce a different approach for computing representations of convoution operators in the waveet basis which consists of soving the system of inear agebraic equations (5.2) subject to asymptotic conditions. This method is especiay simpe if the symbo of the operator is homogeneous of some degree since in this case the operator is competey defined by its representation on V 0. We consider two exampes of such operators, the Hibert transform the operator of fractiona differentiation (or antidifferentiation). The Hibert transform. We appy our method to the computation of the nonstard form of the Hibert transform (5.12) g(x) = (Hf)(y) = 1 π p.v. f(s) s x ds, where p.v. denotes a principa vaue at s = x. The representation of H on V 0 is defined by the coefficients (5.13) r = ϕ(x ) (Hϕ)(x) dx, Z, which, in turn, competey define a other coefficients of the nonstard form. Namey, H = {A j, B j, Γ j } j Z, A j = A 0, B j = B 0, Γ j = Γ 0, where matrix eements α i, β i, γ i of A 0, B 0, Γ 0 are computed from the coefficients r, (5.14) (5.15) α i = β i = k=0 k =0 k=0 k =0 g k g k r 2i+k k, g k h k r 2i+k k,

18 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1733 (5.16) γ i = k=0 k =0 h k g k r 2i+k k. The coefficients r, Z in (5.13) satisfy the foowing system of inear agebraic equations: (5.17) L/2 r = r a 2k 1 (r 2 2k+1 + r 2+2k 1 ), 2 k=1 where the coefficients a 2k 1 are given in (3.19). obtain the asymptotics of r for arge, (5.18) By rewriting (5.13) in terms of ˆϕ(ξ), (5.19) r = 1 π + O ( 1 2M r = 2 0 Using (5.4), (5.6), (5.7), we ). ˆϕ(ξ) 2 sin(ξ) dξ. we obtain r = r set r 0 = 0. We note that the coefficient r 0 cannot be determined from equations (5.17) (5.18). Soving (5.17) with the asymptotic condition (5.18), we compute the coefficients r, 0 with any prescribed accuracy. We note that the generaization for computing the coefficients of Riesz transforms in higher dimensions is straightforward. Exampe. We compute (see Tabe 4) the coefficients r of the Hibert transform for Daubechies waveets with six vanishing moments with accuracy The coefficients for > 16 are obtained using asymptotics (5.18). (We note that r = r r 0 = 0.) Tabe 4 The coefficients r, = 1,, 16 of the Hibert transform for Daubechies waveet with six vanishing moments. Coefficients Coefficients r r M =

19 1734 G. BEYLKIN Fractiona derivatives. We use the foowing definition of fractiona derivatives: (5.20) ( α x f) (x) = (x y) α 1 + Γ( α) f(y)dy, where we consider α 1, 2. If α < 0, then (5.20) defines fractiona antiderivatives. The representation of α x on V 0 is determined by the coefficients (5.21) r = ϕ(x ) ( x α ϕ) (x) dx, Z, provided that this integra exists. The nonstard form α x = {A j, B j, Γ j } j Z is computed via A j = 2 αj A 0, B j = 2 αj B 0, Γ j = 2 αj Γ 0, where matrix eements α i, β i, γ i of A 0, B 0, Γ 0 are obtained from the coefficients r, (5.22) α i = 2 α k=0 k =0 g k g k r 2i+k k, (5.23) β i = 2 α k=0 k =0 g k h k r 2i+k k, (5.24) γ i = 2 α k=0 k =0 h k g k r 2i+k k. It is easy to verify that the coefficients r satisfy the foowing system of inear agebraic equations: r = 2 α r L/2 (5.25) a 2k 1 (r 2 2k+1 + r 2+2k 1 ), 2 k=1 where the coefficients a 2k 1 are given in (3.19). Using (5.4), (5.6), (5.7), we obtain the asymptotics of r for arge, ( ) 1 1 r = Γ( α) 1+α + O 1 (5.26) 1+α+2M for > 0, (5.27) r = 0 for < 0. Exampe. We compute (see Tabe 5) the coefficients r of the fractiona derivative with α = 0.5 for Daubechies waveets with six vanishing moments with accuracy The coefficients for r, > 14, or < 7 are obtained using asymptotics r = 1 ( ) (5.28) + O for > 0, π (5.29) r = 0 for < 0.

20 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1735 Tabe 5 The coefficients {r }, = 7,, 14 of the fractiona derivative α = 0.5 for Daubechies waveet with six vanishing moments. Coefficients Coefficients r r M = E E E E E E E E E E E E E E E E E E E Shift operator on V 0 fast waveet decomposition of a circuant shifts of a vector. Let us consider a shift by one on the subspace V 0 represented by the matrix (6.1) t (0) i j = δ i j,1, where δ is the Kronecker symbo. Using (5.1) with the a n of (3.19) we have (6.2) t (0) = δ,1, t (1) = 1 2 a 2 1,. The ony nonzero coefficients t (j) on each scae j are those with indices L + 2 L 2. Aso, t (j) δ,0 as j. As an exampe, the foowing Tabe 6 contains the coefficients t (j), j = 1, 2,, 8, for the shift operator in Daubechies waveet basis with three vanishing moments. We note that the shift by an integer other than one is treated simiary. However, if the absoute vaue of the shift is greater than L 2, then, on the first severa scaes j, there are nonzero coefficients t (j) with outside the interva L 2. As j increases, a the nonzero coefficients t (j) wi have indices in the interva L 1. The importance of the shift operator stems from the fact that the coefficients of waveet transforms are not shift invariant. However, as we have just demonstrated, the nonstard (, therefore, the stard) forms of the shift operator are sparse easy to compute. By appying these sparse representations directy to the waveet coefficients, in many appications we can effectivey compensate the absence of the shift invariance of the waveet transforms. For exampe, if the representation of a vector in the waveet basis is sparse, there is a corresponding reduction in the number of operations required to shift such a vector. Specificay, in image processing the shift

21 1736 G. BEYLKIN The coefficients {t (j) j = 1,, 8. } =L 2 = L+2 Tabe 6 for Daubechies waveet with three vanishing moments, where L = 6 Coefficients Coefficients t (j) t (j) j = j = E E E E E E E E E E E E-05 j = j = E E E E E E E E E E E E E E-06 j = E-05 j = E E E E E E E E E E E E E E-06

22 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1737 j = E-05 j = E E E E E E E E E E E E E E E-06 operator aows us to move pictures in the compressed form. The coefficients t (j) for the shift operators can be stored in advance used as needed. It is cear, however, that the method of using sparseness of the shift operator depends on the specific appication may be ess straightforward than is indicated above. The foowing is an exampe of an appication where, instead of computing shift operators, we compute a possibe shifts. We describe a fast agorithm for the waveet decomposition of a circuant shifts of a vector then show how it may be used to reduce storage requirements of one of the agorithms of [2]. We reca that the decomposition of a vector of ength N = 2 n into a waveet basis requires O(N) operations. Since the coefficients are not shift invariant, the computation of the waveet expansion of a N circuant shifts of a vector appears to require O(N 2 ) operations. We notice, however, that there are ony N og 2 (N) distinct coefficients present here a simpe agorithm to compute the waveet expansion of N circuant shifts of a vector in N og(n) operations. We reca the pyramid scheme (6.3) {s 0 k } {s1 k } {s2 k } {s3 k } {d 1 k } {d2 k } {d3 k } where the coefficients s 0 k (6.4) for k = 1, 2,, N are given, n= s j k = n=0 h n s j 1 n+2k 1, (6.5) n= d j k = n=0 g n s j 1 n+2k 1, s j k dj k are periodic sequences with the period 2n j, j = 0, 1,, n. In the pyramid scheme (6.3), on each scae j we compute one vector of differences {d j k }k=2n j k=1 one vector of averages {s j k }k=2n j k=1. Instead, et us compute on each scae j, (1 j n), 2 j vectors of differences 2 j vectors of averages. We proceed

23 1738 G. BEYLKIN as foows: et s j 1 k, k = 1,, 2 n j be one of the vectors of averages on the previous scae j 1 compute (6.6) n= s j k (0) = n=0 h n s j 1 n+2k 1, (6.7) n= s j k (1) = n=0 h n s j 1 n+2k, (6.8) n= d j k (0) = n=0 g n s j 1 n+2k 1, (6.9) n= d j k (1) = n=0 g n s j 1 n+2k. To compute the sum in (6.7) (6.9), we shift by one the sequence s j 1 k in (6.6) (6.8). Thus, stepping from scae to scae we doube the number of vectors of averages of differences, at the same time, have the ength of each of them. Therefore, the tota number of operations in this computation is O(N og N). Let us organize the vectors of differences averages as foows: on the first scae, j = 1, we set (6.10) v 1 = (d 1 k(0), d 1 k(1)) (6.11) u 1 = (s 1 k(0), s 1 k(1)), where d 1 k (0), d1 k (1), s1 k (0), s1 k (1) are computed from s0 k On the second scae, j = 2, we set according to (6.6) (6.9). (6.12) v 2 = (d 2 k(00), d 2 k(01), d 2 k(10), d 2 k(11)) (6.13) u 2 = (s 2 k(00), s 2 k(01), s 2 k(10), s 2 k(11)), where d 2 k (00), d2 k (01), s2 k (00), s2 k (01) are computed from s1 k (0) according to (6.6) (6.9) d 2 k (10), d2 k (11), s2 k (10), s2 k (11) from s1 k (1), etc. We caim that we have computed a the coefficients of the waveet expansion of N circuant shifts of the vector s 0 k, k = 1, 2,, N. Indeed, the periodic sequence d 1 k (0) contains a the coefficients that appear if s 0 k is circuanty shifted by 2, 4,, 2n (see (6.8)), the periodic sequence d 1 k (1) contains a the coefficients for odd shifts by 1, 3,, 2 n 1 (see (6.9)). By the same token, the periodic sequences s 1 k (0) s1 k (1) contain a possibe coefficients for even odd circuant shifts of s 0 k (see (6.6) (6.7)). Repeating this procedure on the

24 OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS 1739 next scae for both s 1 k (0) s1 k (1), we again obtain a possibe coefficients for odd even shifts which we coect in v 2 u 2, etc. Whie the vectors v 1, v 2,, v n contain a the coefficients, these coefficients are not organized sequentiay. In order to access them, we generate two tabes i oc (i s, j) i b (i s, j) in O(N og N) operations as foows. For each shift i s, 0 i s N 1 of the vector s 0 k, k = 1, 2,, N, et us write the binary expansion of i s, (6.14) i s = =n 1 =0 ɛ 2, where ɛ = 0, 1. For a fixed scae j, 1 j n, we compute (6.15) i oc (i s, j) = =j 1 =0 ɛ 2, (6.16) i b (i s, j) = =j =n 1 ɛ 2, where i b (i s, j) = 0 if j = n. The number i b (i s, j) points to the begining of the subvector of differences in v j. Namey, the subvector of v j has indices between i b (i s, j) + 1 i b (i s, j) + 2 n j. Within this subvector (which is treated as a periodic vector with the period 2 n j ) the number i oc (i s, j) points to the first eement. For a scaes j, 1 j n, shifts i s, 0 i s N 1, we compute two tabes in (6.15) (6.16). These tabes give us the direct access to the coefficients in vectors v 1, v 2,, v n for a constant cost per eement. We now briefy describe one of the appications of the agorithm for the fast waveet decomposition of a circuant shifts of a vector in numerica anaysis. The agorithms of [2] are designed to evauate the Caderon Zygmund or a pseudodifferentia operator T with kerne K(x, y), (6.17) g(x) = K(x, y) f(y) dy by constructing (for any fixed accuracy) its sparse nonstard or stard form thereby, reducing the cost of appying it to a function. Let us rewrite (6.17) as (6.18) g(x) = K(x, x z) f(x z) dz. If the operator T is a convoution, then K(x, x z) = K(z) is a function of z ony. The nonstard form of a convoution requires at most O(og N) of storage (see the previous section), whie the stard form of [2] wi contain O(N) or O(N og N) significant entries even for a convoution. Aternativey, the stard form of K(x, x z) = K(z) in variabes x z for the convoution operators contains no more than O(og N) significant entries for any fixed accuracy, since the kerne depends on one variabe ony.

25 1740 G. BEYLKIN If we now construct the stard form of K(x, x z) in variabes x z for pseudodifferentia operators (not necessariy convoutions), we obtain super -compression of the operator. Indeed, if these operators are represented in the form (6.18), then the dependence of the kerne K(x, ) on x is smooth the number of significant entries in the stard form is of O(og 2 N). The apparent difficuty in computing via (6.18) is that it is necessary to compute the waveet decomposition of f(x z) for every x thus, it appears to require O(N 2 ) operations. The agorithm of this section accompishes this task in O(N og N) operations. Therefore, the cost of evauating (6.18) does not exceed O(N og N). The advantage of such an agorithm is the reduced storage of O(og 2 N) significant entries for the stard form, which is an important consideration for the mutidimensiona operators. We note that the extension of the agorithm for the fast waveet decomposition of a circuant shifts of a vector to the mutidimensiona case is straightforward. REFERENCES [1] I. Daubechies, Orthonorma bases of compacty supported waveets, Comm. Pure App. Math., 41 (1988), pp [2] G. Beykin, R. Coifman, V. Rokhin, Fast waveet transform numerica agorithms I., Tech. Report YALEU/DCS/RR-696, Yae University, New Haven, August 1989; Comm. Pure App. Math., 44 (1991) pp [3] Y. Meyer, Le cacu scientifique, es ondeettes et fitres miroirs en quadrature, CEREMADE, Université Paris Dauphine, No [4], Ondeettes et functions spines, Tech. Report, Séminaire EDP, Ecoe Poytechnique, Paris, France, [5] S. Maat, Mutiresoution approximation waveets, Tech. Report, GRASP Lab, Dept. of Computer Information Science, University of Pennsyvania, Phiadephia, PA. [6] I. Daubechies J. Lagarias, Two-scae difference equations, I. Goba reguarity of soutions, preprint, Two-scae difference equations, II. Loca reguarity, infinite products of matrices fractas, preprint. [7] A. Cohen, I. Daubechies, J. C. Feauveau, Biorthogona bases of compacty supported waveets, preprint. [8] W. M. Lawton, Necessary sufficient conditions for constructing orthonorma waveet bases, J. Math. Phys., 32 (1991), pp